In previous lectures we have studied electromagnetic waves
travelling in vacuum. We now extend this treatment to propagation in
non-conducting materials. In particular we will be interested in the propagation
of waves from one media into a second different media (reflection and refraction
effects).
Summary
of electromagnetic waves in vacuum
Maxwell's
equations in a vacuum lead to wave equations for E and B. The
resultant waves propagate with a velocity c=(e_{0}m_{0})^{-1/2}. | |
E, B and the propagation direction are mutually perpendicular
(TEM). | |
E and B are in phase and have amplitudes related by E=cB. |
Propagation
in LIH non-conducting media
Propagation
velocity and refractive index
The treatment in this case parallels that in a vacuum except that
we must replace e_{0} by e_{0}e_{r} and m_{0} by m_{0}m_{r}. Hence the speed of light in the medium, c_{m}, is given
by
c_{m}=1/(e_{0}e_{r}m_{0}m_{r})^{1/2} = c/(e_{r}m_{r})^{1/2}
Where c is the speed of light in a vacuum.
However the only materials that have a m_{r}
which is significantly different from 1 are non-LIH ones (e.g. iron). Hence for
most LIH non-conducting materials c_{m}» c/(e_{r})^{1/2}.
The refractive index n of a given material is defined as the speed
of light in vacuum divided by that in the material. Hence
n = c/c_{m} = (e_{r}m_{r})^{1/2} » (e_{r})^{1/2}
or
n^{2} » e_{r}
Because e_{r} always has a value greater than one, the speed of light in a
material is always less than in a vacuum.
As both e_{r} and n may vary strongly with frequency, discrepancies may arise
when comparing values of e_{r} (often the static, DC value is used) with n^{2 }(generally
the value appropriate to optical frequencies).
Relationship
between E and B
In vacuum E=cB.
In a material this is modified to
E=c_{m}B=cB/n
and because
H=B/(m_{0}m_{r})»B/m_{0} as m_{r}»1
Þ H=nE/(cm_{0})
Reflection and refraction at the interface between two different
materials
The aim is to establish the properties of electromagnetic waves
when they encounter a plane interface separating two different non-conducting
materials (generally two different dielectrics). In particular equations will be
derived which give the fractions of the incident wave which are reflected and
transmitted at the interface
Mathematical
description of a plane wave not propagating along one of the principal axes
So far we have only considered waves which propagate along one of
the principal axes. For example a plane wave propagating along the x-axis
is described by an equation of the form E=E_{0}sin(kx-wt),
one propagating along the y-axis by E=E_{0}sin(ky-wt) etc.
In the following we will need to consider plane waves which do
not propagate along one of these principal axes. However we can always
resolve the direction of the wave onto two or more of the principal axes.
For example in the figure below the wave lies in the x/y-plane
and propagates in a direction which makes an angle q to the y-axis. There are hence components cosq along the y-axis and sinq
along the x-axis. The equation for
this plane wave is hence of the form
E=E_{0}sin(k(xsinq+ycosq)-wt)
The wave
vector, k, in the above equations is given by 2p/l where l
is the wavelength of the wave. In addition if the wave propagates in a material
with a velocity c_{m} then
w=c_{m}k=(c/n)k
where w is the angular frequency of the wave. Hence
k=nw/c
Boundary
conditions for E, D,
B and H at an interface
In the lectures on dielectrics
and magnetic materials it was
shown that, in the absence of free surface charge and conduction currents, the
following boundary conditions existed for E,
D, B and H
D
and B: Normal components continuous
E
and H: Tangential components
continuous
These boundary conditions will be used below for electromagnetic waves incident at the boundary between two materials.
Frequency
and direction of reflected and refracted waves
In the figure below a plane electromagnetic wave travelling in a
material of refractive index n_{1} is incident on the plane boundary
with a second material of refractive index n_{2}. The incident wave
propagates at an angle q_{i}
to the normal of the boundary and has E-
and H-fields of magnitude E_{i}
and H_{i}. The E-field
of the incident wave lies in the plane of incidence (the x/y-plane), this is known as the E parallel configuration.
For this configuration the B-
and H-fields must be normal to the
plane of incidence.
In general there will be, in addition to the incident wave, a
reflected wave at an angle q_{r} and a refracted or transmitted wave at an angle q_{t}.
Each of these waves will have E- and H-fields
as shown with amplitudes related by the expression H=nE/m_{0}c where n is the refractive index of the appropriate material.
The E-fields of the three waves are given by the following expressions
E_{i}=E_{i0}sin(k_{1}(xsinq_{i}-ycosq_{i})-w_{1}t) | (A) |
E_{r}=E_{r0}sin(k_{1}(xsinq_{r}+ycosq_{r})-w_{1}t) | (B) |
E_{t}=E_{t0}sin(k_{2}(xsinq_{t}-ycosq_{t})-w_{2}t) | (C) |
Where
E_{i0}, E_{r0} and E_{t0}
are the amplitudes of the E-fields
and k_{1} and k_{2} and w_{1} and w_{2} are the wavevectors and angular frequencies of the waves in the
two materials.
The boundary condition for E
requires that the tangential component (the component parallel to the boundary)
be continuous. Hence if we evaluate this component on both sides of the boundary
we must obtain the same result.
The tangential component of each E-field is given by the magnitude of the E-field multiplied by cosq, where q is the appropriate angle.
On the side of the boundary in material 1 the total tangential
component of the E-field is the
difference of the components due to the incident and reflected waves (see above
figure). In material 2 there is only the transmitted wave.
Hence the requirement that the tangential component of the E-field
be continuous can be written
E_{i}cosq_{i}ç_{y=0 }-
E_{r}cosq_{r}ç_{y=0}=E_{t}cosq_{t}ç_{y=0}
(D)
Where ç_{y=0}
indicates that the preceding expression is evaluated at the boundary y=0.
Substituting in (D) the expressions for E_{i}, E_{r}
and E_{t} given by (A), (B)
and (C) and evaluated for y=0
E_{i0}cosq_{i}sin(k_{1}xsinq_{i}-w_{1}t)-E_{r0}cosq_{r}sin(k_{1}xsinq_{r}-w_{1}t)=E_{t0}cosq_{t}sin(k_{2}xsinq_{t}-w_{2}t)
(E)
This equation must hold at all times and for all values of x.
This is only possible if all the coefficients of x
and all the coefficients of t are
equal. This requires
w_{1}=w_{2} and k_{1}sinq_{i}=k_{1}sinq_{r}=k_{2}sinq_{t}
Hence
The
frequencies of the waves in the two materials are equal w_{1}=w_{2}
| |
The
angles of incidence and reflection are equal k_{1}sinq_{i}=k_{1}sinq_{r}
Þ
q_{i}=q_{r } | |
The
incident and transmitted angles are related by k_{1}sinq_{i}=k_{2}sinq_{t}.
However as k_{1}=n_{1}w_{1}/c and k_{2}=n_{2}w_{2}/c
= k_{1}=n_{2}w_{1}/c
the expression k_{1}sinq_{i}=k_{2}sinq_{t}.
can be written as n_{1}sinq_{i}=n_{2}sinq_{t}.
This is Snell’s law of refraction. |
The above results would have been obtained if E were polarized normal to the plane of incidence (H polarised parallel to the plane of incidence). These results hence apply to any incident wave.
Amplitudes
of the reflected and refracted waves
The above procedure provided information on the frequencies and
directions of the incident, reflected and transmitted waves. The next step is to
derive expressions which give their relative amplitudes.
Returning to equation (E) above, which is valid when E
is parallel to the plane of incidence, the spatial and time dependent
components have been shown to be equal. Hence (E) reduces to
E_{i0}cosq_{i }- E_{r0}cosq_{i}=E_{t0}cosq_{t}
(F)
Where q_{r} has been replaced by q_{i}.
In addition the tangential component of the H-field must be continuous at the boundary between the two
materials. For the present case H is
normal to the incident plane so that the tangential component of H
is simply H.
For the tangential component of H to be continuous we must therefore have
H_{i0}+H_{r0}=H_{t0}
But H=nE/m_{0}c so the above can be written as
n_{1}E_{i0}+n_{1}E_{r0}=n_{2}E_{t0}
(G)
Equations (F) and (G) can now be used to eliminate either E_{r0}
or E_{t0}.
Eliminating E_{t0}:
From
(G) E_{t0}=(E_{i0}+E_{r0})n_{1}/n_{2}
Substituting
into (F)
E_{i0}cosq_{i}-E_{r0}cosq_{i}=(E_{i0}+E_{r0})(n_{1}/n_{2})cosq_{t}
E_{i0}(cosq_{i}-(n_{1}/n_{2})cosq_{t})=E_{r0}(cosq_{i}+(n_{1}/n_{2})cosq_{t})
Multiplying
through by n_{2}
E_{i0}(n_{2}cosq_{i}-n_{1}cosq_{t})=E_{r0}(n_{2}cosq_{i}+n_{1}cosq_{t})
(H)
Now
eliminating E_{r0}. From (G)
E_{r0}=(n_{2}E_{t0}-n_{1}E_{i0})/n_{1}
Substituting
into (F)
E_{i0}cosq_{i}-(n_{2}E_{t0}-n_{1}E_{i0})/n_{1}cosq_{i}=E_{t0}cosq_{t}
E_{i0}n_{1}cosq_{i}-(n_{2}E_{t0}-n_{1}E_{i0})cosq_{i}=E_{t0}n_{1}cosq_{t}
2E_{i0}n_{1}cosq_{i}=E_{t0}(n_{1}cosq_{t}+n_{2}cosq_{i})
(I)
r_{//} and t_{//},
which relate the amplitudes of the reflected and transmitted E-fields
to that of the incident E-field, are
known as the reflection and transmission coefficients for E
parallel to the plane of incidence.
The polarisation of E
can also be aligned normal to the plane of incidence (H is now parallel to the plane of incidence). The boundary
conditions that the tangential components of E and H are continuous
now require:
E_{i0}+E_{r0}=E_{t0}
H_{i0}cosq_{i}-H_{r0}cosq_{i}=H_{t0}cosq_{t}
Again eliminating either E_{r0}
or E_{t0} from these two
equations leads to expressions for the E
perpendicular reflection r_{^} and transmission t_{^} coefficients:
(J)
(K)
The equation (H)-(K) are known as the Fresnel relationships or
equations.
The signs of these equations give the relative phases of the waves. If positive there is no phase change, if negative there is a p phase change.
Properties
of the Fresnel Equations
The properties can be more easily seen if we consider a special
case where one of the materials is air (n=1) and the other has a refractive
index n.
Consider the case where the wave is incident from air. Hence n_{1}=1
and n_{2}=n. The Fresnel equations become
with sinq_{i}/sinq_{t}=n
The above figure (plotted for n=2)
shows a number of points:
For normal incidence (q_{i}=q_{t}=0) both r_{//} and r_{^} and t_{//} and t_{^ }have the same magnitudes
êr_{//}ê=êr_{^}ê=(n-1)/(n+1),
êt_{//}ê=êt_{^}ê=2/(1+n)
As q_{i}®90°
the magnitudes of the reflection coefficients tend to 1 (total reflection) and
the magnitude of the transmission coefficients tend to zero (zero transmission).
For a certain angle q_{B},
known as the Brewster angle, r_{//} becomes zero whereas r_{^} remains non-zero. For this angle
light can only be reflected with E perpendicular to the plane of incidence.
If unpolarised light is incident at the Brewster angle then the
reflected light will be polarised.
The Brewster angle is found by setting r_{//}=0.
Þ
ncosq_{i}=cosq_{t}
(L)
In
addition Snell’s law gives us
nsinq_{t}=sinq_{i}
(M)
dividing (L) by (M)
(ncosq_{i})/(nsinq_{t})=cosq_{i}/sinq_{t}=cosq_{t}/sinq_{i}
Þ cosq_{i}sinq_{i}= cosq_{t}sinq_{t}
and hence
sin2q_{i}=sin2q_{t} as 2sinqcosq=sin2q
The equality sin2q_{i}=sin2q_{t} implies that either q_{i}=q_{t} or q_{i}=90-q_{t}. The former can not
be correct as we also must have nsinq_{t}=sinq_{i} and n¹1.
Hence at the Brewster angle we must have q_{i}=90-q_{t}
and Snell’s law becomes
nsinq_{t}=sinq_{B} nsin(90-q_{B})=sinq_{B}
but sin(90-q)=cosq hence
ncosq_{B}=sinq_{B}
Þ
n=sinq_{B}/cosq_{B}=tanq_{B}
Waves
propagating from a material into air
We now have n_{1}=n and n_{2}=1. Fresnel’s
equations have the form
and nsinq_{i}=sinq_{t}
In this case the equations for r_{//} and r_{^} and for t_{//} and t_{^} are interchanged compared to those for
when the wave is incident from air.
However because nsinq_{i}=sinq_{t} Þq_{t}>q_{i}.
For q_{i} equal to a certain value, the critical angle q_{c},
sinq_{t }becomes equal to unity and hence q_{t}=90°.
At this point both reflection coefficients become equal to one and
the waves are totally reflected. As the value of sinq_{t}
can not exceed unity, for q_{i}>q_{c }the reflection coefficients remain equal to unity.
Hence for q_{i}>q_{c} all the incident light is reflected. This is known as ‘total
internal reflection’.
q_{i}=q_{c} occurs when sinq_{t}=1. Hence nsinq_{c}=1
Þ sinq_{c}=1/n
Power
reflection and transmission coefficients
The reflection and transmission coefficients derived above give the
amplitudes of the electric fields associated with the waves.
However in general it is the power of the wave which is measured
experimentally. From Lecture
19 the power density of an electromagnetic wave is given by the
Poynting vector N=ExH.
This has a magnitude EH = nE^{2}/(cm_{0}),
using H=nE/(cm_{0}).
Hence the energy density µnE^{2}
The coefficients which give the fraction of energy reflected or
transmitted at a boundary between two materials equal the appropriate values of
r^{2} or t^{2} with the inclusion of the appropriate value(s) of
n.
If R is the power
reflection coefficient at an interface between a material of refractive index n_{1}
and one of n_{2} then
where r is the
appropriate reflection coefficient given by the Fresnel relationships.
Similarly if T is the
power transmission coefficient
because energy must always be conserved
R+T=1
Example:
What are the reflection and transmission power coefficients for
light incident normally from air into a material of refractive index n?
q_{i}=q_{t}=0° and ½r_{//}½=½r_{^}½=(n-1)/(n+1)
Conclusions
Propagation
velocity and refractive index | |
Relationship
between E
and B
in a material | |
Mathematical
description of a plane wave not propagating along one of the principal axes | |
Frequency
and direction of reflected and refracted waves | |
Amplitudes
of the reflected and refracted waves: the Fresnel equations | |
Properties
of the Fresnel equations | |
Brewster
angle | |
Critical
angle and total internal reflection | |
Power reflection and transmission coefficients |